Numerical Calculation of the Multiplicity of a Solution to Algebraic Equations

A method to calculate numerically the multiplicity of a solution to a system of algebraic equations is presented. The method is an application of Zeuthen's rule which gives the multiplicity of a solution as the multiplicity of a united point of an algebraic correspondence defined naturally by the system. The numerical calculation is applicable to a large scale system of algebraic equations which may have a solution that we cannot calculate the multiplicity by a symbolic computation.

     

Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications

This paper is concerned with accurate matrix multiplication in floating-point arithmetic. Recently, an accurate summation algorithm was developed by Rump et al. (SIAM J Sci Comput 31(1):189–224, 2008). The key technique of their method is a fast error-free splitting of floating-point numbers. Using this technique, we first develop an error-free transformation of a product of two floating-point matrices into a sum of floating-point matrices. Next, we partially apply this error-free transformation and develop an algorithm which aims to output an accurate approximation of the matrix product. In addition, an a priori error estimate is given. It is a characteristic of the proposed method that in terms of computation as well as in terms of memory consumption, the dominant part of our algorithm is constituted by ordinary floating-point matrix multiplications. The routine for matrix multiplication is highly optimized using BLAS, so that our algorithms show a good computational performance. Although our algorithms require a significant amount of working memory, they are significantly faster than ‘gemmx’ in XBLAS when all sizes of matrices are large enough to realize nearly peak performance of ‘gemm’. Numerical examples illustrate the efficiency of the proposed method.     

Fast algorithms for floating-point interval matrix multiplication

We discuss several methods for real interval matrix multiplication. First, earlier studies of fast algorithms for interval matrix multiplication are introduced: naive interval arithmetic, interval arithmetic by midpoint-radius form by Oishi-Rump and its fast variant by Ogita-Oishi. Next, three new and fast algorithms are developed. The proposed algorithms require one, two or three matrix products, respectively. The point is that our algorithms quickly predict which terms become dominant radii in interval computations. We propose a hybrid method to predict which algorithm is suitable for optimizing performance and width of the result. Numerical examples are presented to show the efficiency of the proposed algorithms.     

Normally contracting Lie group actions

We show that there are no normally contracting actions of unimodular Lie groups on closed manifolds.     

Potentiometric sensor for sulfur oxides using NASICON as a solid electrolyte

The ac conductivity of NASICON is higher by two orders of magnitude than that of Na₂SO₄ at 1000 K. The dc polarization measurement reveals that NASICON shows sodium ion conduction even at the temperature of about 1200 K, and that the electronic transference number is of the order of 10^?5. The SO₂-O₂-SO₃ concentration cell using NASICON electrolyte gives essentially the same electromotive force as in the cell using Na₂SO₄ electrolyte because a thin layer of Na₂SO₄ if formed on NASICON at the electrodes. The high sinterability of NASICON offers a dense electrolyte without permeation of gases. The SO_x sensor using NASICON electrolyte exhibits good response and excellent selectivity against CO₂ and NO₂.     

Selective synthesis of C-arylglycosides via Cp*RuCl-catalyzed partially intramolecular cyclotrimerizations of C-alkynylglycosides

In the presence of catalytic amounts of Cp*RuCl(cod), the partially intramolecular cyclotrimerizations of various C-alkynylglycosides and C-diynylglycosides proceeded at ambient temperature to afford C-arylglycosides.     

Synthesis of benzo-fused lactams and lactones via Ru(II)-catalyzed cycloaddition of amide- and ester-tethered alpha,omega-diynes with terminal alkynes: electronic directing effect of internal conjugated carbonyl group

In the presence of a catalytic amount of Cp*RuCl(cod), 1,6- and 1,7-diynes connected by an amide or an ester tether underwent cycloaddition with terminal alkynes at room temperature to give rise to cycloadducts in 40-93% yields with 63 : 37-83 : 17 regioisomer ratios.     

Acidity effects on the fluorescence properties and adsorptive behavior of rhodamine 6G molecules at the air-water interface studied with confocal fluorescence microscopy

The effects of acidity on fluorescence originated from rhodamine 6G (R6G) molecules adsorbed at the air-water interface of extremely low-concentration aqueous solutions have been studied with confocal fluorescence microscopy. Similarities and differences in the observed acidity effects between R6G molecules at the interface and those in the bulk solution have been discussed. With increasing the subphase-pH from 1 to 6, height and frequency of photon bursts as well as intensity of the interface-originated fluorescence change in two steps, while bulk fluorescence changes in one step and a little change in the number of adsorbed R6G molecules is verified with surface tension measurements. The results suggest that there is an interface-specific equilibrium among the chemical forms of R6G molecules. Chemical forms contributing to the interface-originated fluorescence above pH 5 are discussed.